{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T17:12:09Z","timestamp":1776791529290,"version":"3.51.2"},"reference-count":14,"publisher":"American Mathematical Society (AMS)","issue":"271","license":[{"start":{"date-parts":[[2011,1,13]],"date-time":"2011-01-13T00:00:00Z","timestamp":1294876800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    The goal of this paper is to study some numerical approximations of particular Hamilton-Jacobi-Bellman equations in dimension\u00a01 and with possibly discontinuous initial data. We investigate two anti-diffusive numerical schemes; the first one is based on the Ultra-Bee scheme, and the second one is based on the Fast Marching Method. We prove the convergence and derive\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L Superscript 1\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">L^1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -error estimates for both schemes. We also provide numerical examples to validate their accuracy in solving smooth and discontinuous solutions.\n                  <\/p>","DOI":"10.1090\/s0025-5718-10-02311-2","type":"journal-article","created":{"date-parts":[[2010,4,15]],"date-time":"2010-04-15T09:01:02Z","timestamp":1271322062000},"page":"1395-1426","source":"Crossref","is-referenced-by-count":5,"title":["\ud835\udc3f\u00b9-error estimates for numerical approximations of Hamilton-Jacobi-Bellman equations in dimension 1"],"prefix":"10.1090","volume":"79","author":[{"given":"Olivier","family":"Bokanowski","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nicolas","family":"Forcadel","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hasnaa","family":"Zidani","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2010,1,13]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1023\/A:1007633810484","article-title":"High order numerical discretization for Hamilton-Jacobi equations on triangular meshes","volume":"15","author":"Augoula, Steeve","year":"2000","journal-title":"J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0885-7474","issn-type":"print"},{"key":"2","doi-asserted-by":"crossref","unstructured":"M. Bardi and I. Capuzzo-Dolcetta. Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Systems & Control: Foundations & Applications. Birkh\u00e4user Boston Inc., Boston, MA, 1997. With appendices by Maurizio Falcone and Pierpaolo Soravia.","DOI":"10.1007\/978-0-8176-4755-1"},{"key":"3","series-title":"Math\\'{e}matiques \\& Applications (Berlin) [Mathematics \\& Applications]","isbn-type":"print","volume-title":"Solutions de viscosit\\'{e} des \\'{e}quations de Hamilton-Jacobi","volume":"17","author":"Barles, Guy","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/3540584226"},{"key":"4","doi-asserted-by":"crossref","unstructured":"O. Bokanowski, N. Megdich, and H. Zidani. Convergence of a non-monotone scheme for Hamilton-Jacobi-Bellman equations with discontinuous initial data. To appear in Numerische Math. DOI: 10.1007\/S00211-009-0271-1.","DOI":"10.1007\/s00211-009-0271-1"},{"issue":"1","key":"5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10915-005-9017-0","article-title":"Anti-dissipative schemes for advection and application to Hamilton-Jacobi-Bellmann equations","volume":"30","author":"Bokanowski, Olivier","year":"2007","journal-title":"J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0885-7474","issn-type":"print"},{"issue":"6","key":"6","doi-asserted-by":"publisher","first-page":"1013","DOI":"10.1137\/0721063","article-title":"Averaged multivalued solutions for scalar conservation laws","volume":"21","author":"Brenier, Yann","year":"1984","journal-title":"SIAM J. Numer. 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Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"10","key":"9","doi-asserted-by":"publisher","first-page":"939","DOI":"10.1016\/S0764-4442(99)80301-2","article-title":"Un sch\u00e9ma non lin\u00e9aire anti-dissipatif pour l\u2019\u00e9quation d\u2019advection lin\u00e9aire","volume":"328","author":"Despr\u00e9s, Bruno","year":"1999","journal-title":"C. R. Acad. Sci. Paris S\\'{e}r. I Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-4442","issn-type":"print"},{"issue":"4","key":"10","doi-asserted-by":"publisher","first-page":"479","DOI":"10.1023\/A:1013298408777","article-title":"Contact discontinuity capturing schemes for linear advection and compressible gas dynamics","volume":"16","author":"Despr\u00e9s, Bruno","year":"2001","journal-title":"J. Sci. 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Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"4","key":"13","doi-asserted-by":"publisher","first-page":"907","DOI":"10.1137\/0728049","article-title":"High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations","volume":"28","author":"Osher, Stanley","year":"1991","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"4","key":"14","doi-asserted-by":"publisher","first-page":"1591","DOI":"10.1073\/pnas.93.4.1591","article-title":"A fast marching level set method for monotonically advancing fronts","volume":"93","author":"Sethian, J. A.","year":"1996","journal-title":"Proc. Nat. Acad. Sci. 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